We investigate whether the L¹-algebra of polynomial hypergroups has non-zero bounded point derivations. We show that the existence of such point derivations heavily depends on growth properties of the Haar weights. Many examples are studied in detail. We can thus demonstrate that the L¹-algebras of hypergroups have properties (connected with amenability) that are very different from those of groups.

We consider a continuous derivation D on a Banach algebra 𝓐 such that p(D) is a compact operator for some polynomial p. It is shown that either 𝓐 has a nonzero finite-dimensional ideal not contained in the radical rad(𝓐) of 𝓐 or there exists another polynomial p̃ such that p̃(D) maps 𝓐 into rad(𝓐). A special case where Dⁿ is compact is discussed in greater detail.

In this paper we extend the notion of $n$-weak amenability of a Banach algebra $𝒜$ when $n\in \mathbb{N}$. Technical calculations show that when $𝒜$ is Arens regular or an ideal in ${𝒜}^{**}$, then ${𝒜}^{*}$ is an ${𝒜}^{\left(2n\right)}$-module and this idea leads to a number of interesting results on Banach algebras. We then extend the concept of $n$-weak amenability to $n\in \mathbb{Z}$.

In this paper it is shown that for a Brandt semigroup $S$ over a group $G$ with an arbitrary index set $I$, if $G$ is amenable, then the Banach semigroup algebra ${\ell }^1\left(S\right)$ is pseudo-amenable.